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Fractional Numbers

Updated 2018-02-01

There are two ways to represent fractional numbers, fixed-point and floating-point representations.

In Lab 3, we will be dealing with non integer values.

Fixed-Point

The bits are divided into two sections: whole and fractional. The whole section represents the integer part of the number before the decimal number. The fractional is a sum of the fractional part (with negative exponents).

For example:

1001.0110 is 1B3 + 0B2 + 0B1 + 1B0 + 0B-1 + 1B-1 + 1B-2 + 0B-3, where BN denotes \(\times 2^N\).

The fixed point representation has a predetermined location for the decimal, it is known as the radix point.

For a 8 bit register where the radix point is placed in the center, the precision is 0.0625 and the number can range between 0 and 15.9375. (page 10)

We can have more precision by moving the radix point to give more bits to the decimals, but the trade off is we lose the maximum range.

Verilog

How do we make fixed point in Verilog?

We can declare it as signed or unsigned. With a length long enough for the integer and fractional parts:

reg [15:0] position_z;

Then, when we need to use it, we just need to remember that [15:8] bits are for the integral part, and [7:0] are the fractional part.

Assignment

To assign a value, we just need to split the number into integer and fractional part. For example, to store 2.5, we know the integer part is 10 and the fractional 0.5 part is 10. Assuming our fixed point number is 8 bits long with radix point in the middle, the binary representation is 0010.1000.

Addition & Subtraction

To add the fixed point numbers, we just add the two together (treating them as integers). The overflow (carry-bit) is taken care of automatically.

reg [15:0] position, velocity;
position <= position + velocity;

Subtraction is the same thing except negative.

Multiplication

To multiply, we treat bits as integers and perform multiplication. But we need to shift the radix point and shift accordingly. For example, 000011.01 times 000110.10 gets 101010010. Since the two multiplicand have two decimal places, we need to put 4 bits for the output of the fractional part: 10101.0010. Then we need to fix the output back to the fixed radix by shifting the bits right by 2. Finally, our actual answer is 010101.00 .

Note that we are losing precision when we performed the shift. This is a fundamental disadvantage of fixed-point numbers.

Comparison

left as an exercise

Floating Point

If we care about the precision when the magnitude is small. but not so much when the magnitude is large, we could use floating point.

Consider the number 7241.0381 again. We can represent this as \(72410381\times 10^{-4}\). Thus, the representation follows

\[\text{significant}\times\text{base}^{\text{exponent}}\]

The exponent and significant/mantissa are stored as bits.

Single Precision (32-bit) has 1 sign bit, 8 exponent bits, and the rest are mantissa bits.

Double Precision (64-bit) has 1 sign bit, X exponent bits, and the rest are mantissa bits

Example

50.5625 is 110010.1001 . To represent this in floating point, we need to first find the normalized scientific notion: \(5.05625\times 10^-1\). Note that it is normalized and only has one digit before the decimal place. The binary is now 1.100101001B5 We don’t need to represent the first bit before the decimal. So the reset of the decimals are stored in mantissa. The mantissa now is: 1001010010000000000000.

The exponent is \(2^5\), and 5 is expressed in binary as 101. The exponent bits are signed, but we don’t represent negative in two’s compliment because comparison would be more difficult. The solution is to adding a bias / offset to put the exponent into unsigned range. The bias is a standard.

The bias for single-precision is 127, and 1023 for double precision. This is part of the standard.

So we take 5 (101) and we add 127 (01111111) and we get 10000100 for the exponent bits.

Since the number is positive, the sign bit is 0.

Finally, the whole 32 bit number will look like 0 | 10000100 | 100101001000000000000.

Note that real keyword doesn’t work on FPGAs and cannot be synthesized.

In modern FPGAs, there are floating point blocks that require special instructions to activate. There are libraries available for working with floating point format.

Fixed Point vs. Floating Point

Fixed Point

Floating point